Mathematical modeling of multi-layer polarizers on meander lines

When considering the problem of transformation of electromagnetic field of the linear polarization within the field of elliptic polarization, and vice versa, the multi-layer meander polarizers (MMP) in the wide band are of practical interest. [1] shows the solution of the boundary problem of electrodynamics as well as defines the scattering signatures of a flat wave on one polarizer layer.

For a number of layers however an approximate approach based on neglect of interaction between the layers as per the highest type of spatial harmonics has been used. This allowed to find the scattering matrix of a multi-layer polarizer via the scattering matrix of one of its layers. The problem related to estimate of applicability boundaries of such approach has not been con-sidered.

The objective of this work is the solution of the boundary problem for MMP in the strict electrodynamic setting, taking into account the full interaction between the layers of the polarizers.

Let’s consider several dielectric layers in some of which the meander lines (ML) made of some infinitely fine ideal conductive material are periodically located (Fig.1). Let be one of the sides of the i-th ML surface. The boundary problem of electrodynamics can be formulated as follows: find the second electromagnetic field, the electric vector of which meets the following condition

(1)

it meets the Maxwell’s homogeneous equation outside the location area of its sources; in case of infinity it meets the radiation conditions providing for no-wave coming from the infinity and having no source over there, on ridges — its meets the Meixner conditions [2], and in the corners — the integrability condition. In (1) — is the tangent outside electrical field, — is the tangent component of the secondary electrical field excited by the electric current induced at the MMP conductors taking into account the medium interface.

Fig.1

As is well known [3], the formulated problem has the unique solution. The outside electromagnetic field will be considered to have the form of a monochrome () flat wave. Then by virtue of the MMP geometry the Floquet’s theorem [4] can be used.

Let’s introduce the following symbols: — is the carryover factor of the i-th Floquet’s harmonics (i- is the generalized index [4]) from the l-st into the s-th layer (); — is the “reflection” factor of the i-th Floquet’s harmonics from the upper bound of the l-st layer; — is the “reflection” factor of the i-th Floquet’s harmonics from the bottom bound of the l-st layer.

The and factors (where l<s) can be found when exciting the l-st layer of the multi-layer structure (while all ML are absent) by the Floquet's harmonics going in the positive direction of the OZ axis,
and and factors (where l>s) — by the Floquet’s harmonics going in the negative direction of the OZ axis.

By applying at the appropriate layer the Lorentz lemma towards the ancillary field and the required field in the integrated form [3] and using the orthogonality condition of the subwaves [5], it is possible to get the expressions combining the field development factors with the electric surface current , which presents the total of currents flowing on both sides of the k-th ML [5].

If using the boundary condition (1) we can get the system of the first genus operator equations (as to terminology of [5]) as respects the currents :

(2)

where — is the result of the solution for the problem of the flat excitation wave on the multi-layer structure for the m-th layer, — is the total number of ML, — ist the Green’s electric tensor function of the Maxwell’s equations

— is here — the symbol of dyadic product of vectors, — is the point of observation, — is the source point, , — is the norm of the i-th eigenwave of the l-th layer [5].

For the convenience a flat wave can be recorded in the spherical coordinate frame